The Chain Rule

Consider a system of three gears, where x, y, and z represent the rotational speeds of the first, second, and third gears, respectively.

The first gear turns the second, which turns the third. The system is analyzed using the Chain Rule with a two-stage variable relationship.

The first stage expresses the dependence of the second gear’s rotation on the first, with z as a function of x.

The second stage expresses the dependence of the third gear’s rotation on the second, with y as a function of z.

It follows that the output of the first stage becomes the input of the next, forming a composite function.

A small change in one gear causes changes in the others. To find the instantaneous rate of change in y on x, the ratio of Delta y over Delta x is taken with the limit as Delta x approaches zero. By introducing the intermediate variable, delta z, this ratio is rewritten.

As Delta x approaches zero, Delta z also approaches zero. Then the limit of the product of ratios is expressed as the product of two derivatives, giving the Chain Rule. This gives the derivative of a composite function as the derivative of the outer function multiplied by the inner function’s derivative.